TY - JOUR

T1 - Corrádi and Hajnal's theorem for sparse random graphs

AU - Balogh, József

AU - Lee, Choongbum

AU - Samotij, Wojciech

PY - 2012/1

Y1 - 2012/1

N2 - In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that if p(n) ≫ (log n/n) 1/2, then asymptotically almost surely every subgraph of G(n, p) with minimum degree at least (2/3 + o(1))np contains a triangle packing that covers all but at most O(p -2) vertices. Moreover, the assumption on p is optimal up to the (log n) 1/2 factor and the presence of the set of O(p -2) uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which says that if one imposes certain natural regularity conditions on all three pairs in a balanced 3-partite graph, then this graph contains a perfect triangle packing.

AB - In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that if p(n) ≫ (log n/n) 1/2, then asymptotically almost surely every subgraph of G(n, p) with minimum degree at least (2/3 + o(1))np contains a triangle packing that covers all but at most O(p -2) vertices. Moreover, the assumption on p is optimal up to the (log n) 1/2 factor and the presence of the set of O(p -2) uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which says that if one imposes certain natural regularity conditions on all three pairs in a balanced 3-partite graph, then this graph contains a perfect triangle packing.

UR - http://www.scopus.com/inward/record.url?scp=84859364783&partnerID=8YFLogxK

U2 - 10.1017/S0963548311000642

DO - 10.1017/S0963548311000642

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AN - SCOPUS:84859364783

VL - 21

SP - 23

EP - 55

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 1-2

ER -